TheZ‐eigenvalues of a symmetric tensor and its application to spectral hypergraph theory

SUMMARY In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties ofZ‐eigenvalues of a real symmetric tensor with even order. We first establish that the maximumZ‐eigenvalue function is a continuous and convex function on the symmetric tensor space and so provide formulas of the convex conjugate function andε‐subdifferential of the maximumZ‐eigenvalue function. Consequently, for anmth‐orderN‐dimensional tensor , we show that the normalized eigenspace associated with maximumZ‐eigenvalue function isρth‐order Hölder stable at with . As a by‐product, we also establish that the maximumZ‐eigenvalue function is always at leastρth‐order semismooth at . As an application, we introduce the characteristic tensor of a hypergraph and show that the maximumZ‐eigenvalue function of the associated characteristic tensor provides a natural link for the combinatorial structure and the analytic structure of the underlying hypergraph. Finally, we establish a variational formula for the second largestZ‐eigenvalue for the characteristic tensor of a hypergraph and use it to provide lower bounds for the bipartition width of a hypergraph. Some numerical examples are also provided to show how one can compute the largest/second‐largestZ‐eigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula.